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We start a systematic study of quarter-BPS operators in four-dimensional N = 4 Super Yang-Mills with gauge group SU(N) making use of recently developed tools in conformal field theory. We adapt the technology of embedding space tensor structures in four dimensions to the problem of computing R-symmetry tensor structures, and we use the underlying chiral algebra to obtain the superconformal Ward identities. This allows us to fix the protected part of the four-point correlators, up to few ambiguities. As applications, we use the Lorentzian inversion formula to study the leading order OPE data in the large N supergravity limit and we make contact with the OPE limit of the five-point function of half-BPS operators.

We initiate the study of one-loop gluon amplitudes in AdS space. These amplitudes were recently computed at tree level for a variety of backgrounds of the form $AdS_{d+1} \times S^3$. For concreteness, we compute the one-loop correction to the massless gluon amplitude on $AdS_5\times S^3$, which corresponds to the four-point correlator of the flavor current multiplet in the dual 4d $\mathcal{N}=2$ SCFT. This requires solving a mixing problem that involves tree-level amplitudes of arbitrarily massive Kaluza-Klein modes. The final answer has the same color structure as in flat space but the dependence on Mandelstam variables is more complicated, with logarithms replaced by polygamma functions., v1: 37 pages, 5 figures, v2: minor improvements, published version

We review the recent developments in the study of conformal field theories in generic space time dimensions using the methods of the conformal bootstrap, in its analytic aspect. These techniques are based solely on symmetries, in particular in the analytic structure and in the associativity of the operator product expansion. We focus on two applications of the analytic conformal bootstrap: the study of the $\epsilon$ expansion of the Wilson Fisher model via the introduction of a dispersion relation and the large $N$ expansion of maximally supersymmetric Super Yang Mills theory in four dimensions., Comment: 32 pages, reference added, Contribution to the Open Access Special Issue "Women Physicists in Astrophysics, Cosmology and Particle Physics", Universe, ISSN 2218-1997

We perform a numerical bootstrap study of the mixed correlator system containing the half-BPS operators of dimension two and three in N = 4 Super Yang-Mills. This setup improves on previous works in the literature that only considered single correlators of one or the other operator. We obtain upper bounds on the leading twist in a given representation of the R-symmetry by imposing gaps on the twist of all operators rather than the dimension of a single one. As a result we find a tension between the large N supergravity predictions and the numerical finite N results already at N similar to 100. A few possible solutions are discussed: the extremal spectrum suggests that at large but finite N, in addition to the double trace operators, there exists a second tower of states with smaller dimension. We also obtain new bounds on the dimension of operators which were not accessible with a single correlator setup. Finally we consider bounds on the OPE coefficients of various operators. The results obtained for the OPE coefficient of the lightest scalar singlet show evidences of a two dimensional conformal manifold.

We computed a set of structures which appear in the four-point function of protected operators of dimension two in $\mathcal{N}=4$ Super Yang Mills with $SU(N)$ gauge group, at any order in a large $N$ expansion. They are determined only by leading order CFT data. By focusing on a specific limit, we made connection with the dual supergravity amplitude in flat space, where such structures correspond to iterated $s$-cuts. We made several checks and we conjecture that the same interpretation holds for supergravity amplitudes on $AdS_5 \times S^5$., Comment: 43 pages, Ancillary files contains various results presented in the paper. V2: Corrected typos, added references and modified figure. V3: reformatted for J.Phys. A, accepted version

Journal of physics / A 53(45), 453002 (2020). doi:10.1088/1751-8121/abb0fe, A review of Boundary and defect conformal field theory: open problems and applications, following a workshop held at Chicheley Hall, Buckinghamshire, UK, 7���8 Sept. 2017. We attempt to provide a broad, bird���s-eye view of the latest progress in boundary and defect conformal field theory in various sub-fields of theoretical physics, including the renormalization group, integrability, conformal bootstrap, topological field theory, supersymmetry, holographic duality, and more. We also discuss open questions and promising research directions in each of these sub-fields, and combinations thereof., Published by IOP Publ., Bristol

We perform a numerical bootstrap study of the mixed correlator system containing the half-BPS operators of dimension two and three in $\mathcal N = 4$ Super Yang-Mills. This setup improves on previous works in the literature that only considered single correlators of one or the other operator. We obtain upper bounds on the leading twist in a given representation of the R-symmetry by imposing gaps on the twist of all operators rather than the dimension of a single one. As a result we find a tension between the large $N$ supergravity predictions and the numerical finite $N$ results already at $N\sim 100$. A few possible solutions are discussed: the extremal spectrum suggests that at large but finite $N$, in addition to the double trace operators, there exists a second tower of states with smaller dimension. We also obtain new bounds on the dimension of operators which were not accessible with a single correlator setup. Finally we consider bounds on the OPE coefficients of various operators. The results obtained for the OPE coefficient of the lightest scalar singlet show evidences of a two dimensional conformal manifold., Comment: 40 pages, 12 figures, 4 tables

We consider higher spin operators in weakly coupled gauge conformal field theories. Crossing symmetry of mixed scalar correlators relates different higher spin towers and we study the consequences for the spectrum and structure constants of higher spin operators of different twists. Constraints are obtained to all loops in perturbation theory. The large spin contributions to the structure constants can be resummed into a theory-dependent prefactor times a universal factor, whose structure of poles agrees with the one that would be obtained from a Witten diagram supergravity computation, although only crossing symmetry is assumed. Finally, our results provide an all loop expression for the double null limit of mixed correlators, which is in perfect agreement with the correlator/Wilson loop correspondence., Comment: 26 pages

We present a dispersion relation in conformal field theory which expresses the four point function as an integral over its single discontinuity. Exploiting the analytic properties of the OPE and crossing symmetry of the correlator, we show that in perturbative settings the correlator depends only on the spectrum of the theory, as well as the OPE coefficients of certain low twist operators, and can be reconstructed unambiguously. In contrast to the Lorentzian inversion formula, the validity of the dispersion relation does not assume Regge behavior and is not restricted to the exchange of spinning operators. As an application, the correlator $\langle \phi \phi \phi \phi \rangle$ in $\phi^4$ theory at the Wilson-Fisher fixed point is computed in closed form to order $\epsilon^2$ in the $\epsilon$ expansion., Comment: 19 pages, 2 figures

We propose a method to analytically solve the bootstrap equation for two point functions in boundary CFT. We consider the analytic structure of the correlator in Lorentzian signature and in particular the discontinuity of bulk and boundary conformal blocks to extract CFT data. As an application, the correlator $\langle \phi \phi \rangle$ in $\phi^4$ theory at the Wilson-Fisher fixed point is computed to order $\epsilon^2$ in the $\epsilon$ expansion., Comment: 22 pages, 4 figures, v2: minor improvements, v3: fix a typo

We explore and exploit the relation between non-planar correlators in ${\cal N}=4$ super-Yang-Mills, and higher-genus closed string amplitudes in type IIB string theory. By conformal field theory techniques we construct the genus-one, four-point string amplitude in AdS$_5\times S^5$ in the low-energy expansion, dual to an ${\cal N}=4$ super-Yang-Mills correlator in the 't Hooft limit at order $1/c^2$ in a strong coupling expansion. In the flat space limit, this maps onto the genus-one, four-point scattering amplitude for type II closed strings in ten dimensions. Using this approach we reproduce several results obtained via string perturbation theory. We also demonstrate a novel mechanism to fix subleading terms in the flat space limit of AdS amplitudes by using string/M-theory., 27+12 pages. v2: typos fixed, ref added

Three-dimensional Chern-Simons vector models display an approximate higher spin symmetry in the large $N$ limit. Their single-trace operators consist of a tower of weakly broken currents, as well as a scalar $\sigma$ of approximate twist $1$ or $2$. We study the consequences of crossing symmetry for the four-point correlator of $\sigma$ in a $1/N$ expansion, using analytic bootstrap techniques. To order $1/N$ we show that crossing symmetry fixes the contribution from the tower of currents, providing an alternative derivation of well-known results by Maldacena and Zhiboedov. When $\sigma$ has twist $1$ its OPE receives a contribution from the exchange of $\sigma$ itself with an arbitrary coefficient, due to the existence of a marginal sextic coupling. We develop the machinery to determine the corrections to the OPE data of double-trace operators due to this, and to similar exchanges. This in turns allows us to fix completely the correlator up to three known truncated solutions to crossing. We then proceed to study the problem to order $1/N^2$. We find that crossing implies the appearance of odd-twist double-trace operators, and calculate their OPE coefficients in a large spin expansion. Also, surprisingly, crossing at order $1/N^2$, implies non-trivial $O(1/N)$ anomalous dimensions for even-twist double-trace operators, even though such contributions do not appear in the four-point function at order $1/N$ (in the case where there is no scalar exchange). We argue that this phenomenon arises due to operator mixing. Finally, we analyse the bosonic vector model with a sextic coupling without gauge interactions, and determine the order $1/N^2$ corrections to the dimensions of twist-$2$ double-trace operators., Comment: 19 pages + appendices, v2: minor corrections, corrected statement about mixing

We propose and demonstrate a new use for conformal field theory (CFT) crossing equations in the context of AdS/CFT: the computation of loop amplitudes in AdS, dual to non-planar correlators in holographic CFTs. Loops in AdS are largely unexplored, mostly due to technical difficulties in direct calculations. We revisit this problem, and the dual $1/N$ expansion of CFTs, in two independent ways. The first is to show how to explicitly solve the crossing equations to the first subleading order in $1/N^2$, given a leading order solution. This is done as a systematic expansion in inverse powers of the spin, to all orders. These expansions can be resummed, leading to the CFT data for finite values of the spin. Our second approach involves Mellin space. We show how the polar part of the four-point, loop-level Mellin amplitudes can be fully reconstructed from the leading-order data. The anomalous dimensions computed with both methods agree. In the case of $\phi^4$ theory in AdS, our crossing solution reproduces a previous computation of the one-loop bubble diagram. We can go further, deriving part of the four-point function in $\phi^3+\phi^4$ theory in AdS which had never been computed. In the process, we show how to analytically derive anomalous dimensions from Mellin amplitudes with an infinite series of poles, and discuss applications to more complicated cases such as the ${\cal N}=4$ super-Yang-Mills theory., Comment: 47+12 pages. v3: corrected typos, added reference, discussed subtlety of non-1PI diagrams

Using conformal field theory (CFT) arguments we derive an infinite number of constraints on the large spin expansion of the anomalous dimensions and structure constants of higher spin operators. These arguments rely only on analiticity, unitarity, crossing-symmetry and the structure of the conformal partial wave expansion. We obtain results for both, perturbative CFT to all order in the perturbation parameter, as well as non-perturbatively. For the case of conformal gauge theories this provides a proof of the reciprocity principle to all orders in perturbation theory and provides a new "reciprocity" principle for structure constants. We argue that these results extend also to non-conformal theories., Comment: 20 pages; v2: version published in JHEP

We study the consistency of four-point functions of half-BPS chiral primary operators of weight p in four-dimensional $$ \mathcal{N}=4 $$ superconformal field theories. The resulting conformal bootstrap equations impose non-trivial bounds for the scaling dimension of unprotected local operators transforming in various representations of the R-symmetry group. These bounds generalize recent bounds for operators in the singlet representation, arising from consistency of the four-point function of the stress-energy tensor multiplet.

Made available in DSpace on 2022-04-29T08:30:05Z (GMT). No. of bitstreams: 0 Previous issue date: 2021-07-01 Istituto Nazionale di Fisica Nucleare Ministry of Education, Culture, Sports, Science and Technology This manuscript samples a series of recent results in the quest for a systematic understanding of the space of conformal field theories, with a particular focus on theories with extended supersymmetry. The large majority of results reported here were presented during the second Pollica summer workshop which took place from June 3-21 2019 and focused on mathematical and geometric tools for superconformal field theories. This manuscript represents in many ways a partial summary of the workshop. Mathematical Institute University of Oxford, Woodstock Road University of Cincinnati Physics Department Theoretical Physics Department CERN University of Texas Austin Physics Department C N Yang Institute for Theoretical Physics Stony Brook University Simons Center for Geometry and Physics Stony Brook University Dipartimento di Fisica Ettore Pancini Universita degli Studi di Napoli Federico II, Monte S Angelo, Via Cintia INFN Sezione di Napoli, Monte S Angelo, Via Cintia Department of Physics Princeton University Jadwin Hall Indian Institute of Science Education and Research, Homi Bhabha Rd, Pashan Institut Philippe Meyer Departement de Physique Ecole Normale Superieure Universite PSL Institut de Physique Theorique CEA Saclay Department of Physics and Astronomy Purdue University Purdue Quantum Science and Engineering Institute (PQSEI) Purdue University Perimeter Institute for Theoretical Physics Instituto de Fsica Teorica UNESP ICTP South American Institute for Fundamental Research DESY Theory Group, Notkestraße 85 Theoretical Physics Group The Blackett Laboratory Imperial College London, Prince Consort Road Department of Physics Technion Department of Particle Physics and Astrophysics Weizmann Institute of Science Dipartimento di Fisica Università di Milano-Bicocca INFN Sezione di Milano-Bicocca, Piazza della Scienza 3 Kavli IPMU WPI University of Tokyo, Chiba Niels Bohr Institute Copenhagen University, Blegdamsvej 17 Princeton Center for Theoretical Science Princeton University Department of Physics and Astronomy Uppsala University, Box 516 School of Natural Sciences Institute for Advanced Study, 1 Einstein Drive Instituto de Fsica Teorica UNESP ICTP South American Institute for Fundamental Research